Vector Operations: Finding Coordinates And Lengths

Hey guys! Let's dive into some vector math. We've got two vectors, $\vec{a}$ and $\vec{b}$, and we're going to use them to find the coordinates and lengths of some new vectors. This is a super important concept in algebra and has tons of applications in physics, computer graphics, and a bunch of other fields. So, let's get started!

Understanding the Problem

First off, let's break down what the problem is asking us to do. We're given two vectors: $\vec{a} = -2\vec{i} + 6\vec{j} - 4\vec{k}$ and $\vec{b} = -5\vec{i} + 3\vec{j} + 7\vec{k}$. Notice that these vectors are expressed in terms of their components along the x, y, and z axes, represented by the unit vectors $\vec{i}$, $\vec{j}$, and $\vec{k}$, respectively. This is a pretty standard way to define vectors in 3D space. The key is to remember that these unit vectors are just directions, and the coefficients tell us how much of each direction we have. The goal is to determine the coordinates of new vectors created by a combination of the original vectors, and also find their lengths, which represents the magnitude of the vectors. Specifically, we need to calculate two new vectors, $\vec{c}$ and $\vec{d}$, based on the following operations: $\vec{c} = \vec{a} + 2\vec{b}$ and $\vec{d} = 2\vec{a} + \vec{b}$.

So, essentially, we're going to add, subtract, and multiply vectors by scalars (regular numbers). Vector addition and scalar multiplication are the fundamental operations here, and the ability to do these operations correctly is critical. Scalar multiplication involves multiplying each component of a vector by a scalar. When we add vectors, we add their corresponding components. Vector addition is done component-wise. This means that to add two vectors, you add their respective i, j, and k components separately. For example, if $\vec{u} = u_x\vec{i} + u_y\vec{j} + u_z\vec{k}$ and $\vec{v} = v_x\vec{i} + v_y\vec{j} + v_z\vec{k}$, then $\vec{u} + \vec{v} = (u_x + v_x)\vec{i} + (u_y + v_y)\vec{j} + (u_z + v_z)\vec{k}$. To understand the length or magnitude of a vector is important. The length of a vector $\vec{v} = v_x\vec{i} + v_y\vec{j} + v_z\vec{k}$ is calculated using the formula: $|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$. This is derived from the Pythagorean theorem. Remember, vector operations and understanding their geometric interpretations are the cornerstones of many scientific and engineering fields. Mastering these fundamentals gives you a significant advantage.

Now, let's look at how we can do this step-by-step. Remember that each component of the resulting vectors represents its projection onto the respective axis. This geometric viewpoint is important for visualization. So, are you ready to get started with some vector calculations? Let's begin the exciting journey and solve this mathematical problem together. By the end, you'll be able to confidently handle vector operations.

Calculating Vector $\vec{c} = \vec{a} + 2\vec{b}$

Alright, let's start with the first part of our problem: calculating $\vec{c}$. We know that $\vec{c} = \vec{a} + 2\vec{b}$. This means we need to multiply vector $\vec{b}$ by the scalar 2, and then add the result to vector $\vec{a}$. First, let's find $2\vec{b}$. We do this by multiplying each component of $\vec{b}$ by 2. So, if $\vec{b} = -5\vec{i} + 3\vec{j} + 7\vec{k}$, then $2\vec{b} = 2(-5\vec{i} + 3\vec{j} + 7\vec{k}) = -10\vec{i} + 6\vec{j} + 14\vec{k}$. See? Pretty straightforward! We simply multiply the scalar by each component separately.

Now, to find $\vec{c}$, we add $\vec{a}$ and $2\vec{b}$. We know $\vec{a} = -2\vec{i} + 6\vec{j} - 4\vec{k}$ and we just found that $2\vec{b} = -10\vec{i} + 6\vec{j} + 14\vec{k}$. Adding the corresponding components, we get: $\vec{c} = (-2 - 10)\vec{i} + (6 + 6)\vec{j} + (-4 + 14)\vec{k} = -12\vec{i} + 12\vec{j} + 10\vec{k}$. Therefore, the coordinates of vector $\vec{c}$ are (-12, 12, 10). Awesome, we've found the coordinates of $\vec{c}$! That was not too hard, right? Now, let's calculate the length (magnitude) of $\vec{c}$. Remember, the length of a vector $\vec{c} = c_x\vec{i} + c_y\vec{j} + c_z\vec{k}$ is given by $|\vec{c}| = \sqrt{c_x^2 + c_y^2 + c_z^2}$. In our case, $c_x = -12$, $c_y = 12$, and $c_z = 10$. So, $|\vec{c}| = \sqrt{(-12)^2 + (12)^2 + (10)^2} = \sqrt{144 + 144 + 100} = \sqrt{388}$. This is the length of vector $\vec{c}$. We can simplify this to $2\sqrt{97}$, but it's perfectly fine to leave it as $\sqrt{388}$. So, we've successfully found both the coordinates and the length of vector $\vec{c}$! It is really exciting to see how all the calculations come together, isn't it? Practice is critical here, so try a few more examples. Understanding scalar multiplication and vector addition is so important and that’s what we've done here.

Now, let's summarize what we've learned for vector $\vec{c}$. The vector $\vec{c}$ has coordinates (-12, 12, 10), and its length is $\sqrt{388}$ or $2\sqrt{97}$. We are on a roll, and now let’s solve the next part.

Calculating Vector $\vec{d} = 2\vec{a} + \vec{b}$

Okay, guys, let's move on to the second part of the problem and find vector $\vec{d}$. We know that $\vec{d} = 2\vec{a} + \vec{b}$. This is similar to what we did before, but now we're multiplying vector $\vec{a}$ by the scalar 2, and then adding vector $\vec{b}$.

First, let's find $2\vec{a}$. We multiply each component of $\vec{a}$ by 2. Since $\vec{a} = -2\vec{i} + 6\vec{j} - 4\vec{k}$, then $2\vec{a} = 2(-2\vec{i} + 6\vec{j} - 4\vec{k}) = -4\vec{i} + 12\vec{j} - 8\vec{k}$. Simple enough, right? Scalar multiplication is a piece of cake now. Next, to find $\vec{d}$, we add $2\vec{a}$ and $\vec{b}$. We've just calculated $2\vec{a} = -4\vec{i} + 12\vec{j} - 8\vec{k}$, and we know that $\vec{b} = -5\vec{i} + 3\vec{j} + 7\vec{k}$. Adding the corresponding components gives us: $\vec{d} = (-4 - 5)\vec{i} + (12 + 3)\vec{j} + (-8 + 7)\vec{k} = -9\vec{i} + 15\vec{j} - 1\vec{k}$. Therefore, the coordinates of vector $\vec{d}$ are (-9, 15, -1). We are close to the finish line and we can do it!

Now, let's find the length (magnitude) of vector $\vec{d}$. Remember the formula: $|\vec{d}| = \sqrt{d_x^2 + d_y^2 + d_z^2}$. In our case, $d_x = -9$, $d_y = 15$, and $d_z = -1$. So, $|\vec{d}| = \sqrt{(-9)^2 + (15)^2 + (-1)^2} = \sqrt{81 + 225 + 1} = \sqrt{307}$. This is the length of vector $\vec{d}$. Since 307 is a prime number, we can't simplify this any further. So, we've found both the coordinates and the length of vector $\vec{d}$! See? Not too bad at all! Practice is what matters the most. We've gone through everything step by step. Congratulations, you've successfully completed the calculation of both vectors $\vec{c}$ and $\vec{d}$! Understanding these calculations, step by step, is fundamental to your success. Vector operations are key to understanding more advanced concepts, so pat yourself on the back, you’ve done a great job!

Conclusion

So, there you have it, folks! We've successfully found the coordinates and lengths of both $\vec{c}$ and $\vec{d}$. We've used the principles of scalar multiplication and vector addition to calculate the coordinates, and the magnitude formula to find their lengths. This is a fundamental concept in vector algebra. This demonstrates how vectors behave under different operations and provides a foundation for more complex mathematical concepts.

• Coordinates of $\vec{c}$: (-12, 12, 10)
• Length of $\vec{c}$: $\sqrt{388}$ or $2\sqrt{97}$
• Coordinates of $\vec{d}$: (-9, 15, -1)
• Length of $\vec{d}$: $\sqrt{307}$

Keep practicing these problems and you'll become a pro in no time! Remember to always break down the problem into smaller steps. Make sure to understand the fundamental concepts of vector addition, scalar multiplication and vector length calculation, and you’ll be set for success! Keep up the great work and thanks for following along! And remember to always double-check your calculations to avoid any errors. Hope this was helpful!